A general recursion for integrals involving products of Hermite polynomials and its applications

This study presents the derivation of a recursive formula for integrals of products of $N$ Hermite polynomials, establishing a numerically stable scheme for their accurate evaluation in computer codes. The derivation is notably simple and leverages solely the well-established properties of Hermite polynomials and the method of integration by parts. Importantly, our formulation completely circumvents explicit factorials, thereby preventing potential numerical instabilities and overflows, while facilitating high-precision computations for large indices. These findings are of significant relevance to a variety of areas in physics and mathematics. In particular, they offer an efficient and accurate framework for calculating two- and three-body matrix elements in ab initio simulations of few-body systems under a 1D harmonic confinement using the Configuration Interactions approach. A numerical subroutine implementing the recursive formula is provided as supplemental material.

💡 Research Summary

The paper addresses a long‑standing computational bottleneck in quantum many‑body physics: the evaluation of integrals involving products of Hermite polynomials, which appear in matrix elements of two‑ and three‑body contact interactions for particles confined in a one‑dimensional harmonic trap. Traditional approaches rely on closed‑form expressions derived for specific index sets (e.g., Talmi‑Brody‑Moshinsky expansions) or on symbolic integration. Both routes become impractical for large polynomial degrees because they involve explicit factorials that overflow even in arbitrary‑precision arithmetic and because the algebraic expressions become extremely cumbersome.

The authors derive a completely general recursive formula for the integral

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